Why can't I use a loop-cut on this simple mesh?
You have 2 problems that make Blender unable to know how it could build the edge loop, as it needs quads to create edge loops automatically with ctrl R, and it can't deal with ambiguity:


A face inside your mesh that is not visible in Object mode. You need to select it and delete it with X > Faces
An edge loop coming from the bottom which makes a ngon at its end. You need to select it and delete it with X > Edge Loops


Now you can create a new edge loop.